Related papers: Generalized Sierpi\'nski Numbers

In 1960, Sierpi\'nski proved that there exist infinitely many odd positive integers $k$ such that $k\cdot 2^n+1$ is composite for all positive integers $n$. In this paper, we prove some generalizations of Sierpi\'nski's theorem with $2^n$…

In 1960, W. Sierpinski proved that there are infinitely many positive odd numbers $k$, such that for any positive integer $n$, $k\times2^n+1$ is a composite number. Such numbers are called "Sierpinski numbers". In this study, by using…

Let $b\geq 2$ be an integer. We call an integer $k$ a $b$-Sierpi\'{n}ski number if $\gcd(k+1,b-1)=1$ and $k\cdot b^n+1$ is composite for all positive integers $n$. We similarly call $k$ a $b$-Riesel number if $\gcd(k-1,b-1)=1$ and $k\cdot…

A Sierpinski number is an odd positive integer, k, such that no positive integer of the form k * 2^n + 1 is prime. Similar to a Sierpinski number, a Riesel number is an odd positive integer, k, such that no positive integer of the form k *…

For positive integers $b\geq 2$, $k<b$, and $t$, we say that an integer $k_b^{(t)}$ is a $b$-repdigit if $k_b^{(t)}$ can be expressed as the digit $k$ repeated $t$ times in base-$b$ representation, i.e., $k_b^{(t)} =k(b^t-1)/(b-1)$. In the…

In this paper, we investigate the existence of Sierpi\'{n}ski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpi\'{n}ski numbers and Riesel numbers of the…

For $n\leq 1.5 \cdot 10^{10}$, we have found a total number of 1268 solutions to the Erd\"os-Sierpi\'nski problem finding positive integer solutions of $\sigma(n)=\sigma(n+1)$, where $\sigma(n)$ is the sum of the positive divisors of n. On…

Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We prove that for any positive integer $k$, there is a number $m$ for which the equation $\sigma(x)=m$ has exactly $k$ solutions, settling a conjecture of Sierpi\'nski from…

We investigate the problem of finding integers $k$ such that appending any number of copies of the base-ten digit $d$ to $k$ yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits…

In an earlier paper, we gave an abstract formulation of a theorem of Sierpi\'nski in uncountable commutative groups. In this paper, we prove a result which generalizes the earlier formulation.

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ...…

We show that every countable set of partial bijections from an infinite set to itself can be obtained as a composition of just two such partial bijections. This strengthens a result by Higgins, Howie, Mitchell and Ru\v{s}kuc stating that…

Mariusz Meszka has conjectured that given a prime p=2n+1 and a list L containing n positive integers not exceeding n there exists a near 1-factor in K_p whose list of edge-lengths is L. In this paper we propose a generalization of this…

The inverse central binomial series {equation}S_k(z)=\sum_{n=1}^{\infty}\frac{n^k z^n}{\binom{2n}{n}}{equation} popularized by Ap\'ery and Lehmer is evaluated for positive integers $k$ along with the asymptotic behavior for large $k$. It is…

Let $s\ge 2$ be an integer. Denote by $\mu_s$ the least integer so that every integer $\ell >\mu_s$ is the sum of exactly $s$ integers $>1 $ which are pairwise relatively prime. In 1964, Sierpi\'nski asked a determination of $\mu_s$. Let…

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

The generalized Erd\H{o}s-Straus conjecture, proposed by Wac\l{}aw Sierpi\'{n}ski in 1956, asks whether the Diophantine equation \[ \frac{5}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b,c,d \in…

Let $G \otimes _f H$ denote the Sierpi\'nski product of graphs $G$ and $H$ with respect to the function $f$. The Sierpi\'nski general position number ${\rm gp}{_{\rm S}}(G,H)$ is introduced as the cardinality of a largest general position…

It is known that for an arbitrary positive integer \(n\) the sequence \(S(x^n)=(1^n, 2^n, \ldots)\) is complete, meaning that every sufficiently large integer is a sum of distinct \(n\)th powers of positive integers. We prove that every…

We give a characterization of all pairs $(k,n)$ of positive integers for which the ratio $$ \frac{1^k-2^k+3^k-\dots+(-1)^{n+1} n^k}{1^k-2^k+3^k-\dots+(-1)^{n}(n-1)^k} $$ of two consecutive alternating power sums is an integer.